Re: The reality of sets, on a scale of 1 to 10 (theory of theories)

On 3/24/2025 11:46 AM, Ross Finlayson wrote:

On 03/24/2025 07:24 AM, Jim Burns wrote:

On 3/23/2025 2:23 PM, Ross Finlayson wrote:

Combining a strong mathematical platonism with
a strong mathematical universe hypothesis and
a strong logicist positivism, is there "A Theory",
at all?

>
Yes.
>

The Theory?

>
No.

>
So, you'll aver that there is a theory,
then that there's a greater theory,

I suspect that a theory is greater iff
it has the larger domain.
I ask (without expecting an answer)
if that's what you (RF) mean by 'greater'.
The problem presented by
measuring theories by size.of.domain
is that,
if a first.order.theory has an infinite model,
then it has models of all infinite cardinalities.
By that criterion,
there are finite.modeled theories, and
there are infinite.modeled theories,
the infinite.modeled theories having
considerable vagueness surrounding
what they are theories of.

then that there's a greater theory, and so on, in
all higher orders of theory, in the limit complete?
>
A sort of "continuum limit" of theory?
>
The Theory?

Asking "The Domain?" makes more sense to me.
The standard _domain_ for a great deal of mathematics
is, for one of the ordinals β, the von Neumann universe Vᵦ
⎛ V₀ = {}
⎜ Vᵦ = 𝒫Vᵦ₋₁
⎜ For limit ordinal β
⎝ Vᵦ = ⋃ᵞᑉᵝVᵧ
There are various large.cardinal.axioms which
assert _at least_ the existence of initial ordinal χ
such that
χ has some description or other, which
implies the existence of some domain Vᵪ with
the intended properties.
This scheme, the ordinals, the universes,
seem like what you (RF) are asking about,
except that you want a final limit.domain.
I have no problem with standard mathematics,
and those universes are standard mathematics.
There's definitely
a problem with a _naive_ Final Universe.
However, I don't know that there ISN'T
some work.around.
But a big.enough Vᵪ will be what I need it to be,
so I'm not sure I need work.arounds.
In summary and in conclusion,
domains, yes, the domain, no, at least for now.